<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> The general solution of the homogeneous equation:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y^{\prime \prime}-2 y^{\prime}-3 y=0 \to r^2-2r-3=0 \to r_1=-1, r_2=3 \to y=C_1 e^{-x}+C_2 e^{3x}.
\end{equation*}
</div>
<p class="continuation">Assume a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=A e^{2x}, \quad A~ \textrm{is to be determined.}
\end{equation*}
</div>
<p class="continuation">Substituting the solution into the ODE, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
4 A e^{2 x}-2\cdot 2 A e^{2x}-3 A e^{2 x}=3 e^{2 x} \to -3 A=3 \to A=-1.
\end{equation*}
</div>
<p class="continuation">Thus, one particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=- e^{2 x}
\end{equation*}
</div>
<p class="continuation">And the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=C_1 e^{-x}+C_2 e^{3 x}-e^{2 x}.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_6.html#p-120" class="internal">in-context</a></span>
</body>
</html>
